The Game's Afoot, Watson!

Calculus Level 5

The multiple integral $\int_0^\pi \int_0^\pi \int_0^\pi \frac{du\,dv\,dw}{1 - \cos u \cos v \cos w}$ can be shown to be equal to $\tfrac{P}{Q} \Gamma\big(\tfrac{1}{R}\big)^S$ for positive integers $$P,Q,R,S$$, where $$P,Q$$ are coprime. What is the value of $$P+Q+R+S$$?

 Notation: $$\Gamma(\cdot)$$ denotes the Gamma function.

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